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The geometry of Bloch space in the context of quantum random access codes

We study the communication protocol known as a Quantum Random Access Code (QRAC) which encodes $n$ classical bits into $m$ qubits ($m<n$) with a probability of recovering any of the initial $n$ bits of at least $p>\tfrac{1}{2}$. Such a code is denoted by $(n,m,p)$-QRAC. If cooperation is allowed through a shared random string we call it a QRAC with shared randomness. We prove that for any $(n,m,p)$-QRAC with shared randomness the parameter $p$ is upper bounded by $ \tfrac{1}{2}+\tfrac{1}{2}\sqrt{\tfrac{2^{m-1}}{n}}$. For $m=2$ this gives a new bound of $p\le \tfrac{1}{2}+\tfrac{1}{\sqrt{2n}}$ confirming a conjecture by Imamichi and Raymond (AQIS'18). Our bound implies that the previously known analytical constructions of $(3,2,\tfrac{1}{2}+\tfrac{1}{\sqrt{6}})$- , $(4,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{2}})$- and $(6,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{3}})$-QRACs are optimal. To obtain our bound we investigate the geometry of quantum states in the Bloch vector representation and make use of a geometric interpretation of the fact that any two quantum states have a non-negative overlap.